ECO 120 Macroeconomics Week 5

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ECO 120 MacroeconomicsWeek 5
Investment and Savings Lecturer Dr. Rod Duncan
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Topics

• A firms investment decision
• Present value of 1
• Net present value in the investment decision
• Investment demand

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Why are we studying investment?

• Investment (I) is a component of aggregate
  expenditure
• AE C I G NX
• Changes in I will cause changes in AE.
• But any changes in the AE curve, will cause a
  shift in the aggregate demand (AD) curve.
• So any changes in I will lead to a shift in the
  AD curve.

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Investment

• Investment can refer to the purchase of new goods
  that are used for future production. Investment
  can come in the form of machines, buildings,
  roads or bridges. This is called physical
  capital.
• Another type of investment is called human
  capital. This is investment in education,
  training and job skills.
• Usually when we talk about investment, we mean
  investment in physical capital, but investment
  should include all forms of capital.

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Investment decision-making

• What determines investment?
• Businesses or individuals make an investment if
  they expect the investment to be profitable.
• Imagine we have a small business owner who is
  faced with an investment decision.
• The small business owner will make the investment
  as long as the investment is profitable.
• How to determine profitability of investment?

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Profitability of an investment

• Example
• An investment involves the current cost of
  investment (I).
• The investment will pay off with some flow of
  expected future profits.
• The future stream of profits is R1 in one years
  time, R2 in two years time, up to Rn at the
  nth year when the investment ends.
• Imagine you are the business owner. How do we
  decide whether to make the investment? Can we
  simply add up the benefits (profits) and subtract
  the costs (investment)?
• Profits today R1 R2 Rn I?
• What is wrong with this calculation?

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Present value concept

• Imagine our rule about future values was simply
  to add future costs and benefits to costs and
  benefits today.
• Scenario A friend offers you a deal
• Give me 10 today, and I promise to give you 20
  in 1 years time.
• If we subtract costs (10) from benefits (20),
  we get a positive value of 10. Does this seem
  like a sensible decision?
• Scenario A friend offers you a deal
• Give me 10 today, and I promise to give you 20
  in 100 years time.
• If we subtract costs (10) from benefits (20),
  we get a positive value of 10. Does this seem
  like a sensible decision?

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Present value concept

• Not really. The problem is that a 1 today is
  not the same as a 1 in a years time or 100
  years time.
• We can not directly add these 1s together since
  they are not the same things. We are adding
  apples and oranges.
• We need a way of translating future 1s into 1s
  today, so that we can add the benefits and costs
  together.
• The conversion is called present value.
• In making the decision about our friends deal,
  we would compare 10 today to the present value
  of the 20 in a year or 100 years.

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Present value concept

• An investment is about giving up something today
  in order to get back something in the future.
• So an investment decision will always involve
  comparing 1s today to 1s in the future.
• Investment decisions will always involve present
  values. If we subtract the present value of
  future profits from costs today, we get net
  present value.
• Net Present Value (NPV) Present Value of Future
  Profits (PV) Investment (I)

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Net present value

• The investment rule will be to invest if and only
  if
• NPV 0
• Or
• Present Value of Future Profits (PV) Investment
  (I) 0

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Interest rates

• To measure present value we will have to use
  interest rates.
• Interest rates are a general term for the
  percentage return on a dollar for a year
• that you earn from banks for saving
• that you pay banks for borrowing or investing
• For example, the interest rate might be 10, so
  if you put 1 in the bank this year, it will
  become (1i) in one years time.
• Or if you borrow 100 today, you will have to
  repay (1i)100 next year.

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Interest Rates
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Discounting future values

• What is the PV of 1 in a year? How do we place
  a value today on 1 in t years time?
• This is called discounting the future value.
  One way to think about this question is to ask
• How much would we have to put in the bank now to
  have 1 in t years time?
• Money in the bank earns interest at the rate at
  the rate i, igt0. If I put 1 in the bank today,
  it will grow according to the rate of interest.
• We can construct a chart of our bank account over
  time.

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Bank account
Year Value i.10
0 1 1
1 1(1i) 1.10
2 1(1i)(1i) 1.21
3 1(1i)3 1.33

n 1(1i)n (1.1)n

• If we start with 1 in our bank account, what
  happens to our bank account over time?

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How much is a future 1?

• In order to have 1 next year, we would have to
  put x in today
• 1 (1 i) x
• x 1/(1i) lt 1
• 1 next year is worth 1/(1 i) today. Since
  igt0, 1 next year is worth less than 1 today.
• In order to have 1 in n years time, we would
  have to put x in today
• x 1/(1i)n (1i)-n
• 1 in n years time is worth 1/(1i)n lt 1 today.

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PV of 1
Year i0.01 i0.05 i0.10 i0.20
0 1 1 1 1
1 0.99 0.95 0.91 0.83
2 0.98 0.91 0.83 0.69
3 0.97 0.86 0.75 0.58
10 0.91 0.61 0.39 0.16
n (1.01)-n (1.05)-n (1.10)-n (1.20)-n
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Investment decision

• Imagine we are the small business owner we were
  discussing before. We have a new project which
  we might invest in
• An investment involves the current cost of
  investment (I).
• The investment will pay off with some flow of
  expected future profits.
• The future stream of profits is R1 in one years
  time, R2 in two years time, up to Rn at the
  nth year when the investment ends.

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Investment decision
Year Benefit Cost PV
0 0 I -I
1 R1 0 R1/(1i)
2 R2 0 R2/(1i)2
3 R3 0 R3/(1i)3

n Rn 0 Rn/(1i)n
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Net present value

• The NPV of the investment is the sum of the
  values in the far-right column- the PVs.
• NPV R1/(1i) R2/(1 i)2 Rn/(1 i)n I
• If NPV 0, then go ahead and make the
  investment. If NPV lt 0, then the investment is
  not worthwhile.
• Lets look at a more concrete example that we can
  put some numbers to.

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Example of NPV

• Example A small business in Bathurst that owns
  photo store is considering installing a
  state-of-the-art developing machine for digital
  photographs.
• Cost 12,000 (after selling current machine)
• Future benefits 2,000 per year in extra
  business every year for 10 year life-span of
  machine (assume benefits start next year)

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Example of NPV
Year Benefit Cost PV
0 0 I -12,000
1 2,000 0 2,000/(1i)
2 2,000 0 2,000/(1i)2
3 2,000 0 2,000/(1i)3

10 2,000 0 2,000/(1i)10
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Example of NPV

• NPV -12,000 2,000/(1i) 2,000/(1i)2
  2,000/(1i)3 2,000/(1i)10
• Our NPV then depends upon the interest rate, i,
  facing the small business.
• For a small business, the relevant interest rate
  would be the rate that it cost raise the money,
  say by taking out a bank loan.
• So the interest rate would be the bank small
  business loan rate.

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Example of NPV

• The NPV varies with the interest rate
• At i0.05, NPV 3,443, so go ahead with
  investment.
• At i0.08, NPV 1,420, so go ahead with
  investment.
• At i0.10, NPV 289, so go ahead with
  investment.
• At i0.12, NPV -700, so dont go ahead with
  the investment.
• Somewhere between a 10 and a 12 interest rate,
  NPV 0. NPV lt 0 for all interest rates greater
  than 12.

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Example of NPV

• Another way of thinking about this problem is to
  ask Can I repay the loan and still make money?
• The small business owner borrows 12,000 from the
  bank and uses the 2,000 in extra business each
  year to repay the loan.
• Would the business owner repay the loan before
  the machine needs to be replaced?

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Example of NPV- bank loan
Year 0.05 0.08 0.1 0.12
0 -12000 -12000 -12000 -12000
1 -10600 -10960 -11200 -11440
2 -9130 -9836.8 -10320 -10812.8
3 -7586.5 -8623.74 -9352 -10110.3
4 -5965.83 -7313.64 -8287.2 -9323.58
5 -4264.12 -5898.74 -7115.92 -8442.41
6 -2477.32 -4370.63 -5827.51 -7455.49
7 -601.19 -2720.28 -4410.26 -6350.15
8 1368.75 -937.91 -2851.29 -5112.17
9 3437.19 987.06 -1136.42 -3725.63
10 5609.05 3066.03 749.94 -2172.71
Present Value 3443.47 1420.16 289.13 -699.55
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Example of a NPV- bank loan

• So for interest rates of 10 and below, the bank
  loan is repaid before the machine wears out, so
  the investment is worthwhile.
• For interest rates of 12 and above, the bank
  loan is not repaid by the time the machine needs
  to be replaced, so the investment is not
  worthwhile.
• The bottom line shows that the remainder in the
  bank account at the end of 10 years is the NPV of
  the investment decision.
• So another way to think of NPV is as the money
  left in an account at the end of a project.

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Investment demand

• Instead of thinking about a single small
  business, think of a whole economy of businesses
  and individuals making investment decisions.
• Some of these investment decisions will be very
  good ones and some will be very poor ones. There
  is a whole range.
• As i rises, the PV of future profits will drop,
  so the NPV will fall. If we imagine that there
  are thousands of potential investments to be
  made, as i rises, fewer of these potential
  investments will be profitable, and so investment
  will fall.

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Investment demand

• If we graphed the investment demand for goods and
  services (I) against interest rates, it would be
  downward-sloping in i. The higher is i, the
  lower is investment demand.
• What can shift the I curve? Factors that affect
  current and expected future profitability of
  projects
• New technology
• Business expectations
• Business taxes and regulation

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Shifts in investment demand

• Example An increase in business
  confidence/expectations raises the expected
  future profits for businesses.
• At the same interest rates as before, since the
  Rs are higher, the NPVs of all investment
  projects will be higher.
• The investment demand curve is shifted to the
  right. I is higher for all interest rates.

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Uses of PV concept

• Housing valuation We can use the PV concept to
  estimate what house prices should be.
• What do you have when you own a home? You have
  the future housing services of that home plus the
  right to sell the home.
• Value of housing services should be the price
  people pay to rent an equivalent home. Rent is
  the price of a week of housing services.
• Lets say your home rents for 250 per week.

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Housing valuation

• If you stayed in your home for 50 years, your
  house is worth the PV of 50 years of 52 weekly
  250 payments plus any sale value at 50 years.
  How do we calculate the PV of such a long stream
  of numbers?
• Trick For very long streams, the sum
• PV (250 x 52) (250 x 52)/(1i)
• Is very close to
• PV (250 x 52) / i 13,000 / i

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Housing valuation

• So we get the house values
• At i0.02, PV House 650,000
• At i0.03, PV House 433,000
• At i0.05, PV House 260,000
• At i0.06, PV House 217,000
• At i0.07, PV House 186,000
• At a house price above this price, you are better
  off selling your house and renting for 50 years.
  At a house price below this price, you are better
  off owning a house.

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Housing valuation

• You can also see how sensitive house prices are
  to the interest rate. When i rose from 6 to 7,
  the value of the house dropped 31,000.
• You can see why home owners care so much about
  the home loans rates.
• But what about the resale price at 50 years?
• The PV of the house sale in 50 years time is
  (Sale Price) / (1i)50, which for most values of
  i is going to be a very small number- 8 of Sale
  Price at 5 interest and 3 of Sale Price at 7
  interest.

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Housing price bubbles

• Sometimes the price of housing can vary from this
  PV of housing services price. Some analysts
  argue that todays housing prices is one case-
  these periods are called bubbles.
• Example At 6 interest rates our house was
  worth 217,000. Lets say Sam bought the house
  for 300,000 in order to sell the house one year
  from now.
• In order to be able to repay the 300,000, Sam
  has to gain 18,000 (6 of 300,000) by holding
  the house for a year.

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Housing price bubbles

• Since Sam gets 13,000 worth of housing services
  from the house, the value of the house has to
  rise 5,000 to 305,000 in next years sale for a
  total gain of 18,000.
• Even though the house is unchanged, the
  overpayment for the house has to rise- the
  house is still only worth 217,000 in housing
  services- but it now sells for 305,000.
• So in a bubble, if people are overpaying for a
  house, the overpayment has to keep rising.
  Eventually people realize that the house only
  generates 217,000 in services.

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Housing price bubbles

• Example In Holland in 1636, the price of some
  rare and exotic tulip bulbs rose to the
  equivalent of a price of an expensive house.
  People paid that much in plans to resell at even
  higher prices.
• In 1637, prices for tulips crashed and by 1639,
  tulip bulbs were selling for 1/200th of the peak
  prices.
• Bubbles tend to crash fast and dramatically.